Rigidity of Quasi-Einstein Metrics

By Jeffrey S. Case, Yujen Shu, Guofang Wei. 2008

281 citations recorded by the source corpus.

We call a metric quasi-Einstein if the $m$-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some K\"ahler quasi-Einstein metrics.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/51d863886155a29348757cb321f9ad2d57de5524

How researchers see this paper

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In Applied Mathematics

1. Michele Rimoldi
2. G. Catino
3. A. G. Setti
4. S. Pigola
5. M. Rigoli
6. Chenxu He
7. Peter Petersen
8. W. Wylie

In Astronomy and Astrophysics

1. Michele Rimoldi
2. G. Catino
3. A. G. Setti
4. S. Pigola
5. M. Rigoli
6. Chenxu He
7. Peter Petersen
8. W. Wylie

In Geometry and Topology

1. Michele Rimoldi
2. A. G. Setti
3. S. Pigola
4. M. Rigoli
5. G. Catino
6. Chenxu He
7. Peter Petersen
8. W. Wylie

Citations in the corpus

• A remark on Einstein warped products (2010, methodology): “The proof of Theorem 3 will require the following formula obtained in [Case et al. 2011], which generalizes to the case m < +∞ similar formulas for Ricci solitons (m = +∞) obtained previously by Petersen and Wylie [2009].”
• On Bach flat warped product Einstein manifolds (2011, unknown): “12) in [Case et al. 2011], and (2-7) was shown in [He et al.”
• Generalized quasi-Einstein manifolds with harmonic Weyl tensor (2010, unknown): “Natural examples of GQE manifolds are given by Einstein manifolds (when f and λ are two constants), gradient Ricci solitons (when λ is constant and μ = 0), gradient Ricci almost solitons (when μ = 0, see [11]) and quasi-Einstein manifolds (when μ and λ are two constants, see [3,5,9]).”
• On the geometry of gradient einstein-type manifolds (2014, background): “ are, for instance, Ricci almost solitons ([41]), Yamabe solitons ([19], [31],[21], [9]), Yamabe quasi-solitons ([27], [45]), conformal gradient solitons ([43], [15]), quasi-Einstein manifolds ([30], [12], [16], [26], [35]), ρ-Einstein solitons ([17], [18]). In this paper we study Riemannian manifol”
• Ricci almost solitons (2010, background): “Wei introduce in [1] the concept of a “quasi Einstein manifold”, i.”
• On the classification of warped product Einstein metrics (2010, methodology): “ing result from [CSW] is a classification of K¨ahler (λ,n+m)-Einstein metrics for finite m. Their arguments also give a classification when ∂M6= ∅, see Remark 3.8 below. Other extensions of results from [CSW] are discussed in section 5. WARPED PRODUCT EINSTEIN MANIFOLDS 3 The main part of this paper is”